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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 102550r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
102550.r2 | 102550r1 | \([1, 1, 1, -12975213, 17984116531]\) | \(15461341616731869233929/8803814950000\) | \(137559608593750000\) | \([]\) | \(2695680\) | \(2.6145\) | \(\Gamma_0(N)\)-optimal |
102550.r1 | 102550r2 | \([1, 1, 1, -15575588, 10261295781]\) | \(26744657491710624461689/12562375000000000000\) | \(196287109375000000000000\) | \([]\) | \(8087040\) | \(3.1638\) |
Rank
sage: E.rank()
The elliptic curves in class 102550r have rank \(0\).
Complex multiplication
The elliptic curves in class 102550r do not have complex multiplication.Modular form 102550.2.a.r
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.